Cheating or no?

So, I never thought I’d need to ask but here I am. So, my man got super drunk last night, and for the first time in my entire life, I went on his phone while he was asleep. I just had a gut feeling I needed to check if he was cheating. I go on and see a girl texting him asking him to come over to have lunch and she keeps trying to get him to come over. He keeps making excuses why he can’t instead of really just saying “I have a girlfriend” because it was clear this girl really wanted him over. And my boyfriend replies with the long “sorryyyy” and “I’m busyyyy” which for him is a flirting behavior. She then texts him a few days ago “what are you doing tomorrow” and he texted “working why”, she doesn’t reply. Then he texts her hours later “why what’s up” and then AGAIN at midnight of the same day texts her again “okay turd 😒” which again, flirting behavior for him. Why does he want to know what she wants so bad if she didn’t reply? Why doesn’t he make it clear he has a girlfriend? He has other female friends that he doesn’t ever talk to in this manner, he talks like that to me and me only, or so I thought.

I know I broke a lot of trust boundaries going on his phone, and I need a concrete point to bring it up without seeming crazy.